AI for Everyone

Chapter 12

Uniform & Normal Distribution

Uniform distribution spreads probability evenly over an interval; normal distribution is bell-shaped around the mean. Used in AI for initialization, noise, and priors.

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Uniform & Normal Distribution

Uniform distribution spreads probability evenly over an interval; normal distribution is bell-shaped around the mean. Used in AI for initialization, noise, and priors.

Uniform & Normal distribution

Uniform and normal are two of the most used continuous distributions. Their shape is determined by mean and variance (Ch10–Ch11).
Uniform distribution — Same height over an interval [a,b][a,b]. Density f(x)=1/(b−a)f(x) = 1/(b-a) for a≤x≤ba \le x \le b. Used when outcomes are equally likely (e.g. one face of a die).
Uniform mean is (a+b)/2(a+b)/2, variance is (b−a)2/12(b-a)^2/12. The center of the interval is the mean; wider intervals give larger variance.
Normal distribution — Determined by mean μ\mu and standard deviation σ\sigma. Density f(x)=1σ2π e−(x−μ)2/(2σ2)f(x) = \frac{1}{\sigma\sqrt{2\pi}}\,e^{-(x-\mu)^2/(2\sigma^2)}. Fits measurement error, height, scores (values cluster near the mean).
Bell curve — Normal is highest at the mean and tapers off on both sides. Symmetric about μ\mu; about 68% in μ±σ\mu \pm \sigma, about 95% in μ±2σ\mu \pm 2\sigma.
Why these two? — Uniform is used when we have no prior information (initialization, flat prior). Normal appears for noise/error and via the central limit theorem (averages tend to be normal), so both are central in AI and statistics.
Priors — In Bayesian settings, uniform is a common 'uninformative' prior; normal is used when we have beliefs about mean and variance.
Noise and error — Regression errors, VAE and diffusion noise are often modeled as normal. The math is simple and matches the central limit idea.
Central limit theorem — With many independent trials, the sample mean tends to a normal distribution. So confidence intervals and hypothesis tests rely on normality.
In deep learning and ML — Weight initialization (uniform/normal), dropout and noise (normal), VAE latent space (normal), diffusion (Gaussian) all use these distributions.
Initialization — Weights are drawn from uniform or normal. Too large or biased values hurt training; usually small-variance normal is used.
Noise — VAE samples the latent vector from a normal; diffusion models add and remove Gaussian noise step by step.
Regression — Assuming normal errors makes least squares (OLS) equivalent to maximum likelihood. Prediction intervals use μ±kσ\mu \pm k\sigma.
Bayesian — Uniform or normal priors are common; after observing data we compute the posterior. Neural network weights can have normal priors.
Math flow — Ch10 random variables and distributions, Ch11 mean and variance, then Ch12 two concrete distributions (uniform and normal). Knowing these helps read 'initialization', 'noise', and 'prior' in AI papers.
Uniform — On [a,b][a,b], density 1/(b−a)1/(b-a), mean (a+b)/2(a+b)/2, variance (b−a)2/12(b-a)^2/12. Normal — Mean μ\mu, variance σ2\sigma^2; interval probabilities from standard normal table or calculator.
Example (uniform). On [0,6][0,6], mean is 33, variance 36/12=336/12=3, standard deviation 3\sqrt{3}.
Example (normal). For mean 7070 and standard deviation 1010, about 68% lie in 6060–8080, about 95% in 5050–9090.